Tight Bounds on the Complexity of Recognizing Odd-Ranked Elements

Let S = s1, s2, s3, ..., sn be a given vector of n distinct real numbers. The rank of z R with respect to S is defined as the number of elements si S such that si z. We consider the following decision problem: determine whether the odd-numbered elements s1, s3, s5, . . . are precisely the elements of S whose rank with respect to S is odd. We prove a bound of (n log n) on the number of operations required to solve this problem in the algebraic computation tree model. Let S = s1, s2, s3, . . . , sn Rn be a given vector. For an arbitrary real z, define the rank of z with respect to S, denoted by rankS(z), as the number of elements of S less than or equal to z. Thus, for instance, the largest element of S has rank n. Let odd(S) denote the set of elements of S whose rank with respect to S is odd. We consider the following problem: given a vector S = s1, s2, s3, ..., sn of n distinct real numbers, determine whether the odd-numbered elements s1, s3, s5, . . . are precisely the elements of...

• Type: Journal
• Year: 2006
• Where: CORR
• Authors: Shripad Thite
• Keywords: Algebraic Computation Tree, CORR 2006, Distinct Real Numbers, Education, Odd-numbered Elements